Prog. Theor. Phys. Vol. 44 No. 3 (1970) pp. 689-702
Clebsch-Gordan Formulas of the SU(1, 1) Group
Department of Physics, Tohoku University, Sendai
(Received April 28, 1970)
The Clebsch-Gordan formulas for d-function are studied for both nonunitary finite dimensional representation and unitary infinite dimensional representation of the SU(1, 1) group. It is shown that, in addition to the usual Clebsch-Gordan formulas for the above-mentioned two representations, there exists an additional type of Clebsch-Gordan formula connecting the nonunitary and unitary d-functions. The new kind of Clebsch-Gordan coefficient of the SU(1, 1) obtained in our previous paper arises naturally in this formula. The three Clebsch-Gordan formulas are proved to be interrelated with each other by analytic continuation. It is, further, pointed out that all the recurrence formulas of the unitary d-functions –obtainable from the known recurrence relations of Jacobi polynomials– belong actually to this new type of Clebsch-Gordan formula.
Explicit algebraic formulas of d-functions are derived by employing an elementary algebraic method analogous to the standard treatment of angular momentum.
DOI : 10.1143/PTP.44.689
- L. C. Biedenharn, “Group Theoretical Approaches to Nuclear Spectroscopy”, Chap. V in Lecture on Theoretical Physics, ed. W. E. Britten (Interscience, New York, 1963), Vol. V.
H. Ui, Ann. of Phys. 49 (1968), 69[CrossRef].
- For example, J. Strathdee, J. F. Boyce, R. Delbourgo and A. Salam, “Partial Wave Analysis”, Chap. 6, IC/67/9 (Lecture Notes at International Center for Theoretical Physics, Trieste).
- V. Bargmann, Ann. Math. 48 (1947), 568.
- L. C. Biedenharn, J. Nuyts and N. Straumann, Ann. Inst. Henri Poincare 3 (1965), A13.
L. C. Biedenharn, Lecture on Theoretical Physics, Cargese Summer School, Corsica, 1965.
- Bateman Manuscript Project, Higher Transcendental Functions (McGraw-Hill, New York, 1954), Vol. I, II, III.
W. J. Holman III and L. C. Biedenharn, Ann. of Phys. 39 (1966), 1[CrossRef];
ibid. 47 (1968), 205[CrossRef].
- H. Ui, the succeeding paper, Prog. Theor. Phys. 44 (1970), 703[PTP].
Citing Article(s) :
Progress of Theoretical Physics Vol. 69 No. 4 (1983) pp. 1146-1153
A Class of Simple Hamiltonians with Degenerate Ground State. I
Gyo Takeda and Haruo Ui
Progress of Theoretical Physics Vol. 72 No. 2 (1984) pp. 266-284
Does Accidental Degeneracy Imply a Symmetry Group?
Haruo Ui and Gyo Takeda